Digital Filter – Introduction

What is a system?

A system is a rule that maps an input signal to an output signal. In continuous time and discrete time we depict and denote: \[ x(t)\ \longrightarrow\ y(t),\qquad x[n]\ \longrightarrow\ y[n]. \] These are the standard input–output representations used throughout the text.

Input–output operator notation

We often write the system as an operator ( {} ): \[ y(t)=\mathcal{T}\{x(t)\},\qquad y[n]=\mathcal{T}\{x[n]\}. \] Block diagrams are used to represent systems and interconnections (series/cascade and parallel).

Overview: what is an input–output relation?

A system specifies how an input signal produces an output signal: \[ x(t)\ \longrightarrow\ y(t),\qquad x[n]\ \longrightarrow\ y[n]. \] We study types of relations used in analysis and design: - Memoryless (static) mappings - Differential/difference-equation descriptions - Convolution (LTI) - Transform-domain forms (frequency, Laplace, (z)) - State–space (first-order vector form) - Time-varying vs time-invariant, linear vs nonlinear

Memoryless (static) relations

A memoryless or static system relates input and output at the same instant: \[ y(t)=F\!\big(x(t)\big),\qquad y[n]=F\!\big(x[n]\big). \] Examples: (y=|x|), (y=x^2), saturation and clipping nonlinearities. These are common as pointwise nonlinear stages preceding or following LTI blocks.

Linear constant-coefficient differential equations (CT)

Many continuous-time LTI systems are described by an LCCDE: \[ \sum_{k=0}^{N} a_k\,\frac{d^{\,k} y(t)}{dt^{\,k}} \;=\; \sum_{m=0}^{M} b_m\,\frac{d^{\,m} x(t)}{dt^{\,m}}, \qquad a_0\neq 0. \] Coefficients (a_k,b_m) are constants for time invariance. This form covers standard electrical/mechanical systems and filters.

Linear constant-coefficient difference equations (DT)

Discrete-time LTI systems admit an LCCD equation: \[ \sum_{k=0}^{N} a_k\,y[n-k]\;=\;\sum_{m=0}^{M} b_m\,x[n-m],\qquad a_0\neq 0. \] This representation includes IIR/FIR digital filters and many algorithmic recursions.

Convolution relations (LTI)

For LTI systems the input–output relation is convolution with the impulse response: \[ y(t)=\int_{-\infty}^{\infty} h(\tau)\,x(t-\tau)\,d\tau,\qquad y[n]=\sum_{k=-\infty}^{\infty} h[k]\;x[n-k]. \] Here (h(t)) or (h[n]) is the output to a unit impulse. Causality for LTI corresponds to (h(t)=0) for (t<0) (or (h[n]=0) for (n<0)).

Frequency-domain input–output (Fourier)

When Fourier transforms exist, convolution becomes multiplication: \[ Y(\omega)=H(\omega)\,X(\omega),\qquad Y\!\big(e^{j\omega}\big)=H\!\big(e^{j\omega}\big)\,X\!\big(e^{j\omega}\big). \] (H) is the frequency response (CTFT/DTFT). Magnitude (|H|) scales, phase (H) shifts/warps timing.

Laplace and (z)-transform forms

With Laplace and (z)-transforms (for appropriate regions of convergence): \[ Y(s)=H(s)\,X(s),\qquad Y(z)=H(z)\,X(z), \] and for LCCDE/LCCD systems \[ H(s)=\frac{b_0+b_1 s+\cdots+b_M s^M}{a_0+a_1 s+\cdots+a_N s^N},\qquad H(z)=\frac{b_0+b_1 z^{-1}+\cdots+b_M z^{-M}}{a_0+a_1 z^{-1}+\cdots+a_N z^{-N}}. \] These rational forms support pole–zero analysis and stability checks.

State–space (first-order vector) description

An equivalent input–output formulation uses state variables: \[ \dot{\mathbf{s}}(t)=\mathbf{A}\,\mathbf{s}(t)+\mathbf{b}\,x(t),\qquad y(t)=\mathbf{c}^\top\mathbf{s}(t)+d\,x(t). \] Discrete time: \[ \mathbf{s}[n+1]=\mathbf{A}\,\mathbf{s}[n]+\mathbf{b}\,x[n],\qquad y[n]=\mathbf{c}^\top\mathbf{s}[n]+d\,x[n]. \] This first-order form is algebraically equivalent to LCCDE/LCCD for LTI systems.

Time-varying vs time-invariant relations

• Time-invariant (TI): coefficients and operators do not change with time/index; shifts commute with the system.
• Time-varying (TV): coefficients depend on (t) or (n): \[ \sum_{k=0}^{N} a_k(t)\,\frac{d^{\,k} y(t)}{dt^{\,k}} =\sum_{m=0}^{M} b_m(t)\,\frac{d^{\,m} x(t)}{dt^{\,m}}. \] TV models arise in modulated systems and adaptive filtering.

Linear vs nonlinear relations

• Linear: superposition holds — additivity and homogeneity.
• Nonlinear: violates superposition; examples include squares, rectifiers, saturators, quantizers. Nonlinear stages are often analyzed locally or via small-signal linearization around an operating point.

Stochastic input–output (LTI summary)

For wide-sense stationary (WSS) inputs to an LTI system: \[ \mu_y=\mu_x\,H(0)\ \text{(when defined)},\qquad S_{yy}(\omega)=\big|H(\omega)\big|^2\,S_{xx}(\omega). \] This links input and output statistics through (H), supporting noise and SNR analysis.

Interconnections (examples)

• Series (cascade): output of System 1 feeds System 2.
• Parallel: both systems process the same input; outputs are summed. These patterns are foundational for building complex systems from simpler ones.

Property: Memoryless vs. with memory

• Memoryless: the output at an instant depends only on the input at that same instant.
• With memory: the output depends on past/future values (e.g., accumulators, averagers). Example: the summer/accumulator (running sum) and its inverse (first difference) illustrate systems with memory: \[ y[n]=\sum_{k=-\infty}^{n} x[k],\qquad \text{inverse: } y[n]=x[n]-x[n-1]. \] Both are causal (see next slide)

Property: Causality (general and LTI)

Causal: output depends only on present/past input values. For LTI systems, causality is characterized by the impulse response: \[ \text{Discrete time: } h[n]=0\ \text{for } n<0;\qquad \text{Continuous time: } h(t)=0\ \text{for } t<0. \] Under these conditions, the convolution reduces to depend only on past/present input.

Property: Stability (BIBO)

Bounded-Input Bounded-Output (BIBO) stability: bounded input implies bounded output. For LTI systems: \[ \sum_{k=-\infty}^{\infty} |h[k]| < \infty \quad \Longleftrightarrow \quad \text{discrete-time LTI is stable}, \] \[ \int_{-\infty}^{\infty} |h(t)|\,dt < \infty \quad \Longleftrightarrow \quad \text{continuous-time LTI is stable}. \] These are necessary and sufficient conditions.

Property: Linearity (superposition)

A system is linear if it satisfies additivity and homogeneity: for any signals (x_1,x_2) and scalar (c), \[ \mathcal{T}\{x_1+x_2\}=\mathcal{T}\{x_1\}+\mathcal{T}\{x_2\},\qquad \mathcal{T}\{c\,x\}=c\,\mathcal{T}\{x\}. \] (These two conditions together are equivalent to linearity.)

Property: Time invariance (definition)

A system is time-invariant if a shift in the input produces an identical shift in the output: \[ \mathcal{T}\{x(t-t_0)\}=y(t-t_0),\ \text{whenever}\ y(t)=\mathcal{T}\{x(t)\}. \] Analogously in discrete time with shifts by integer indices. (Definition used throughout the text in system properties and LTI analysis.)

Property: Invertibility

A system is invertible if distinct inputs produce distinct outputs; equivalently, there exists an inverse system that recovers the input from the output. Example pair (discrete time): the accumulator and the first-difference operator are inverses: \[ y[n]=\sum_{k=-\infty}^{n} x[k] \quad \Longleftrightarrow \quad x[n]=y[n]-y[n-1]. \]

Additional note: Interconnections and LTI analysis

For LTI systems, interconnections admit simple algebraic descriptions via transforms: e.g., in the Laplace domain, series and parallel lead to product and sum of system functions, respectively: \[ H_{\text{series}}(s)=H_1(s)H_2(s),\qquad H_{\text{parallel}}(s)=H_1(s)+H_2(s). \]

Digital Filter – Introduction

Digital Filter – Introduction

Convolution